Cooperative Learning and Statistics Instruction

Gerald Giraud
University of Nebraska, Lincoln

Journal of Statistics Education v.5, n.3 (1997)

Copyright (c) 1997 by Gerald Giraud, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.


Key Words: Achievement; Attitude; Undergraduates.

Abstract

This study examined the relative effects of cooperative vs. lecture methods of instruction. Two sections of an undergraduate statistics course were studied. Test scores were dependent variables. Students in one section were randomly assigned to cooperative groups. Students in both sections completed assignments and practice problems -- in the cooperative class in groups during class, and in the lecture class individually, outside of class. Students in the cooperative learning class achieved higher test scores. Implications of the study and resulting questions are discussed.

1. Introduction

1 Cooperative learning has been advocated as an instructional methodology because of its effect on achievement and on other attributes that accompany the acquisition of knowledge, including motivation, classroom socialization, the student's confidence in learning, and attitude toward the subject being learned (e.g., Johnson and Johnson 1985, 1986a, 1986b). Cooperative learning has been defined as the instructional use of small groups such that students work together to maximize their own and each other's learning (e.g., Johnson, Johnson, and Smith 1991). In this study, cooperative learning involved students working together in small groups to solve problems and complete assignments in a post-secondary introductory statistics course.

2 Cooperative learning as an instructional method is advocated by the National Council of Teachers of Mathematics (1989, 1991), and the National Research Council (1989). Although these reports focus on elementary and secondary education, arguments for the benefits of cooperative learning apply to the college classroom as well (Garfield 1993; Johnson, Johnson, and Smith 1991).

3 Cooperative learning -- small-group interaction centered on material to be learned -- should enhance learning because this method is logically connected to effective learning strategies as described in the literature on cognition and learning. These strategies include self monitoring and testing, repeated and variable contact with the material to be learned, external connections between the material and outside ideas, and practice with feedback (Pressley 1995).

4 Cooperative learning groups offer opportunities to use all of these strategies. Students' understanding is monitored in comparison to others in the small-group discussions; students read, hear lecture, discuss, and hear others discuss the material, offering a variety of encoding opportunities. Thus, the number and variety of contacts with the material is increased over lecture or independent study. Discussions among students create external connections with the material by reference to a variety of experiences and perspectives; group assignments and cooperative problem solving give an opportunity to practice skills and get immediate feedback from other group members.

5 Cooperative methods also provide the opportunity for more competent students to scaffold tasks as they interact with less competent students. Scaffolding occurs when learners are assisted by others in constructing knowledge (Wood, Bruner, and Ross 1976). Vygotsky's theory of proximal development (1978), which poses that learners gain knowledge through interaction with more knowledgeable others, suggests that cooperative learning groups would advance the acquisition of knowledge for less competent students when they are in groups with more competent students. Thus, students who are disadvantaged in terms of aptitude or prior knowledge for a given domain of study can benefit from contact with students who are more skilled. Cooperative groups provide a natural setting for peer tutoring, and for skilled students to model for less skilled students.

6 Garfield (1993) advocates cooperative learning methods for statistics instruction in college classes, but few studies address the use of cooperative learning in college statistics courses. The objective of this study was to determine the effect of cooperative learning groups on the performance of undergraduate students enrolled in an applied statistics course.

2. Cooperative Learning in College Statistics Classes

7 Statistics instructors have studied the effects of cooperative learning compared to more traditional lecture instruction. Jones (1991), in a study that compared students in statistics classes taught using lecture methods to students taught using cooperative methods, reported an increase in student-teacher interaction, number of office visits, attendance, and students' rating of instruction in classes taught with cooperative methods. Jones reported that students in the cooperative learning class expressed more positive attitudes toward statistics than students in the lecture classes. Jones compared classes in previous years to the most recent class, did not control for differences in student aptitude that might impact statistics performance, and could not compare achievement because the domain taught was expanded in the cooperative class.

8 Keeler and Steinhorst (1995) compared cooperative statistics classes to a class taught with lecture and individual methods in a previous semester, using successful course completion and letter grades as the dependent variables. They found that more students successfully completed the cooperative class, and that those who completed the cooperative class obtained higher marks than students who completed the lecture class. Student aptitude that might impact performance was not considered.

9 Prior studies of cooperative learning methods in statistics courses at the college level have not compared cooperative learning classrooms to control classes taught in the same semester by the same instructor. The current study compared classes taught by the same instructor in the same semester, using the same text, sample problems, and assignments. Previous studies have compared cooperative learning to lecture instruction using variables that might not be directly comparable (letter grades and student statements). The current study compares the students in a cooperative learning class to those in a lecture class in terms of their scores on identical tests.

10 Therefore, in light of previous research and the importance of evaluating a potentially effective means of teaching statistics, a study was designed to test the effectiveness of the cooperative group learning method as applied in a basic and simple form to statistics instruction. The study is designed to answer the following research questions.

  1. Do students in a cooperative learning statistics class obtain higher scores on classroom tests than students in a traditional (lecture) classroom, controlling for instructor and statistics readiness level?
  2. Is there an interaction between students' basic algebra and mathematics skill level and instructional method? Does cooperative learning benefit less skilled students?

3. Method

3.1 Participants

11 Participants for the study were students enrolled in two sections of Educational Psychology 459, an introductory applied statistics course at the University of Nebraska, Lincoln. Class sizes were 44 for the cooperative learning class and 51 for the lecture class. Students did not know at the time of course enrollment that one section would be lecture and the other would incorporate cooperative learning. The classes did not differ significantly in terms of gender composition, self-reported grade point average, hours worked per week, or Statistics Readiness Test (SRT) scores (see Table 1).


Table 1. Demographic Characteristics

                   Male  Female     GPA     Hours worked      SRT
                                   M    SD    M      SD     M     SD

      Cooperative   19     25    3.33  .70  15.10  12.60  20.62  4.98

      Lecture       19     32    3.15  .68  13.63  11.80  19.84  5.53

3.2 Instruments

12 All students enrolled in EP 459 are required to take a 30-question Statistics Readiness Test during the first week of class. The test consists of multiple choice questions constructed to test knowledge of basic algebra and mathematical reasoning. Scores on this test were used as a measure of statistics readiness. The test's title (Statistics Readiness Test) is somewhat misleading: it measures basic mathematics and algebra understanding. This test is used as a measure of statistics readiness on the assumption that an understanding of algebra and mathematical reasoning indicates a student's preparedness to succeed in the the study of statistics. (See the Appendix for sample items.)

13 Four classroom tests were given during the course. Because of differences between sections in administration of Tests 1 and 3, Test 2 and the final examination were used to measure achievement. Test 2 consisted of 29 multiple choice items assessing knowledge of central tendency, dispersion, correlation, and simple regression. Test 4, the final, was a 28-item comprehensive test, with a focus on the last section of instruction, including statistical error, power, and chi-square tests. This test included both multiple choice items and a constructed response item worth two points, for a total value of 29 points. These tests were reviewed by the faculty supervisor for this course prior to administration to students, as well as by other instructors of introductory statistics courses. These reviewers agreed that the tests contained items that reflected the content and level of understanding appropriate for the course. (See the Appendix for sample items.)

3.3 Treatment

14 Various strategies have been suggested for organizing and managing cooperative learning groups. Strategies have been put forward based on research done with elementary and secondary students (e.g., Fantuzzo, King, and Heller 1992; Johnson and Johnson 1975; Slavin 1985). Methods include the use of rewards, structured processes involving periodic tests, summing scores for peer groups, assigning roles to group members, and training students in group processes and interactive skills. Johnson and Johnson (1985, pp. 252-253) pose four conditions for successful group work: 1) interdependence, so that tasks require cooperation, and rewards are such that all members will participate; 2) face-to-face interaction, i.e., groups are small enough for personal interaction; 3) individual accountability, i.e., students are individually accountable for asking for and giving help; and 4) students know or are taught interpersonal and small-group skills.

15 Garfield (1993) suggests that there is no one right way to configure cooperative groups in statistics classes. Citing Johnson, Johnson, and Smith (1991), Garfield lists ways that groups can be formed, including self-selection or assignment by instructor for homogeneity or heterogeneity. She stresses the importance of groups working together to solve problems, analyze data, and discuss the material being learned.

16 Although some researchers have advocated highly structured cooperative group methods, such structure requires time out of instruction in the classroom. In addition, reward structures and complex grading strategies require planning time that might otherwise be better used, especially by a university instructor. The value of an instructional strategy to college students and instructors is found in its ease of use and its effectiveness for them in a natural classroom setting. In this study, I was interested in whether assigning students to cooperative groups, with minimal special structures or planning, could affect student achievement as measured by usual classroom tests.

17 Cooperative learning in this study was organized in a simple, uncomplicated way. At the start of the semester, students in the cooperative learning class were assigned randomly by the instructor to one of nine groups of five students, which remained the same throughout the semester. Random assignment was used to construct groups with a range of ability, to create the opportunity for scaffolding (Vygotsky 1978). Students were told that from time to time during the semester, they would work in groups during class time. Students were informed of basic group skills: everyone should participate, more outgoing students should seek to include less outgoing students, everyone in the group should understand the work of the day and be able to do it. During class meetings when the groups worked together, students rearranged desk-chairs into groups within the classroom. Students were not given special training or assigned group roles, as some strategies for cooperative learning suggest.

18 Both the lecture and the cooperative class met twice a week, for a class period of 75 minutes. In the cooperative class, groups met at least once a week, and often twice a week, for a minimum of 1/2 hour of class time, and occasionally groups met for the entire class period.

19 As topics of the statistics course were presented, both sections were given practice assignments: the lecture class to do individually outside of class, and the cooperative class to work through in groups during class. Students in both sections were responsible for handing in five of the assignments for individual credit; the remaining assignments were for practice only in the cooperative class. To encourage students in the lecture class to complete the practice assignments, they were given extra credit if they went to a tutoring lab and completed the practice assignment. Most students in the lecture class took advantage of this opportunity. Assignments were the same for both groups. Class time not devoted to group work in the lecture class was devoted to reviewing the practice problems and other examples in a lecture/class participation format. While the cooperative class worked on assignments in groups, the instructor observed, answered questions, and informed the class of any insights gained from circulating among the groups.

20 Students in the cooperative class were given class time to meet in groups to discuss impending tests and to work together to compose study sheets for the tests, which were allowed in both classes. In the lecture class, time was devoted to pre-test review in a lecture/class participation format. Conceptual questions about the statistics covered in class were included for discussion in the cooperative class. Conceptual questions were presented in lecture class and discussed by the instructor, with limited student participation, such as questions and brief statements.

21 Rewards for individuals in the cooperative class were implicit: students who were attentive and who cooperated with their groups were able to complete all homework in class. They could ask questions of other group members and compare their problem-solving approaches, and they had an opportunity to compare test readiness with others.

4. Results

22 Achievement was measured using classroom test scores. Because Test 1 was early in the semester, and Test 3 was administered in a different form for each class, analysis was conducted on Test 2 and the comprehensive final examination (Test 4). Observed means and standard deviations for Test 2 and Test 4 are presented in Table 2.


Table 2. Means and Standard Deviations for Tests 2 and 4*, by Instructional Method

               Cooperative      Lecture
                  n = 44         n = 48
                 M      SD      M      SD

      Test 2  25.14    2.51  23.78    4.20

      Test 4  24.65**  2.69  22.31**  3.23


      *  There were 29 items on test 2 and 28 items on test 4.
      ** Adjusted means, using pretest score as a covariate.

23 The algebra pretest (Statistics Readiness Test) score was employed as a covariate. This test score was significantly correlated with student scores on Test 2 (r = .589, p < .001) and Test 4 (r = .32, p = .006). For Test 2, the assumption of homogeneity of regression slopes for analysis of covariance was not met; an analysis of variance for instructional method by four levels of readiness revealed interaction between method and readiness (F(1,65) = 5.84, p < .01). Post hoc comparisons suggest that students who were at the lowest algebra and mathematics skill level (as measured by the pretest) and were in the cooperative learning group scored higher than their counterparts in the lecture class. Observed means for Test 2, by instructional method and pretest level, are presented in Table 3.


Table 3. Means and Standard Deviations for Test 2, by Instruction and Pretest Level (29 Items on Test)

                                      Pretest level
                     1 (low)         2            3         4 (high)
                     M     SD     M     SD     M     SD     M     SD

      Cooperative     n = 7        n = 10       n = 12       n = 7
                   25.00  2.38  24.30  2.71  25.00  1.95  27.57  0.78

      Lecture         n = 12       n = 9        n = 10       n = 6
                   20.16  4.01  26.55  1.24  24.10  3.32  27.50  0.55

24 Using the algebra pretest score as a covariate, the adjusted mean for the cooperative learning class was significantly higher on the final examination (Test 4) (F(1,69) = 11, p = .001). Adjusted mean scores were 24.65 for the cooperative learning class and 22.31 for the lecture class.

25 A comparison of instructional method by pretest level for the final examination indicates that students at each pretest level in the cooperative class scored higher than students at the same level who were in the lecture class (see Table 4).


Table 4. Means and Standard Deviations of Test 4 (Final), by Instruction and Pretest Level (28 Items on Test)

                                  Pretest level
                    1 (low)       2          3       4 (high)
                     M    SD    M    SD    M    SD    M    SD

      Cooperative    n = 7      n = 10     n = 12     n = 7
                   24.6  2.1  23.2  2.5  25.3  1.7  26.7  1.8

      Lecture        n = 11     n = 9      n = 10     n = 6
                   22.7  2.4  20.6  4.3  23.1  1.7  24.0  2.6


5. Discussion

5.1 Benefits of Cooperative Learning

26 As instructor, I observed differences between the two sections. Students in the cooperative learning class, when working on assignments in groups, asked questions more often than they, or the lecture class students, did during lecture, even though I frequently asked for questions during lecture. I observed that the randomly assigned groups worked more closely together as the semester progressed. Students responded positively when I asked them if the small groups helped them understand the material. I felt I was able to gauge the understanding of the cooperative learning students by listening to them discuss problems and assignments, and therefore I felt better able to plan lectures appropriately. I was less confident of my knowledge of the learning status of the lecture class students. I used the insight gained from the small groups to plan lectures for both sections.

27 In terms of achievement, cooperative learning was more effective than lecture and individual work for students who scored lowest on the Statistics Readiness Test, and equally effective for students at higher levels of readiness. Differences in test scores were substantively, as well as statistically, significant. Mean differences on Test 2 for students who scored lowest on the Statistics Readiness Test represented a letter grade difference.

28 Because the cooperative groups were randomly assigned, students who were less prepared for statistics, according to the Statistics Readiness Test, were likely to be in groups with well-prepared students. This contact might explain the significantly higher scores of the lowest statistics readiness level students in the cooperative learning class when compared to the lecture class. This result supports Vygotsky's (1978) concept of scaffolding for less able learners.

29 The lack of difference on Test 2 between students at higher levels of statistics readiness, both within and between treatments, might indicate that a certain threshold of basic algebra knowledge is sufficient to see students through the initial instruction of an introductory statistics course. Test 2 tested central tendency, dispersion, correlations, and simple regression, the first formula-based instruction of the course. Students with a necessary level of basic algebra preparation might be expected to out-perform those who lack it. It is important to note that students at these higher levels of readiness seem not to be harmed, in terms of test scores, by cooperative work.

30 The comprehensive final examination (Test 4) demonstrated the benefit of cooperative learning for students at all statistics readiness levels: the adjusted mean for the cooperative class was higher than that of the lecture class using readiness as a covariate. This result might be attributable to the cumulative effect of the more able students helping the less prepared students understand the more complex material covered late in the course. That is, as the scaffold builders help the climbers, both become stronger in their knowledge of the material. This effect would logically be more pronounced in testing over more difficult material: Test 4 covered material that required more conceptual understanding of how the statistical ideas learned earlier fit together in practice.

31 The effect of cooperative learning on student-instructor interaction might also explain differences in test scores. Students seemed less inhibited about asking questions in the small groups. As one student said, "It's easier to ask a question when it's not in front of the whole class." When students did ask questions, they gained closer interaction with the instructor and immediate feedback on performance.

32 Students, asked for an end-of-course evaluation of cooperative learning, expressed positive attitudes toward cooperative learning: "I think it was a great idea because it allowed people to learn both by teaching others, and by being shown how to do something themselves. Learning in the small groups was easy." "Helpful learning experience ... able to bounce ideas off of group members ... helpful when unsure of material, can ask questions and get a different explanation than comes from the teacher. Overall, very worthwhile and helpful." A few students felt that the cooperative groups allowed some students to take advantage of others: "For our group, one member was not prepared at all." One student felt the group work was not helpful: "When I got in the group to discuss the problems I started to get confused because everybody had a different answer."

5.2 Limitations

33 This was a quasi-experimental study, with several potentially confounding variables and a design restriction that limits the generalizability of results. The design included only one class of each type of instruction. Therefore, between class variation in the dependent variables that might have resulted had cooperative learning and lecture been compared in multiple classrooms cannot be accounted for. The results of this study are limited to the students who were in these two specific classes. Therefore, generalizations to larger populations are tenuous.

34 A possible explanation for differences in student performance is that students in the cooperative class benefited from doing the homework and practice assignments in class, and that students in the lecture class spent less time outside of class on the work than the cooperative class did in class. Differences in performance, then, might be an artifact of time spent on the material, and not a result of cooperative learning groups.

35 Environment and time of day might also have affected results. The cooperative learning section met twice a week in the morning, and the lecture group met in the late afternoon. The cooperative learning section met in a small classroom near a loading dock, and several students commented on the distraction of noisy carts going in and out of the loading dock doors, which were a few feet from the classroom door. This classroom had individual chair/desks, and was oddly shaped, so that one row of students had difficulty seeing parts of the chalkboard. The lecture section met in a larger room where students sat in cushioned chairs behind long tables. This room had a raised platform at the front, and was in an out-of-the-way location, where distractions outside of the room were few. Time of day was not controlled, and it might have been a factor in student response to instruction. However, some students in the morning section commented in an end-of-course evaluation that the class should have been later, while students in the afternoon class wanted an earlier time.

6. Conclusion

36 This study supports the hypothesis that cooperative learning, in the form of randomly assigning students to small groups to work together on a variety of assignments, results in higher achievement than lecture instruction for all students, and is especially beneficial for those least prepared for statistics. This is important for instructors of statistics, as well as for students: instructors need a strategy that serves all students well. Findings also suggest that cooperative learning promotes retention of material for all students, as evidenced by differences in final examination scores.

37 The treatment in this study was relatively simple: students were randomly assigned to cooperative groups at the start of the course, where they remained throughout. Assignments were not specially designed for small groups, but were equally usable by both lecture and cooperative classes. Neither complex evaluative techniques nor an elaborate reward structure was utilized. Students were responsible for their own work and were evaluated individually by traditional testing. This study supports the efficacy of cooperative learning without an extra commitment of instructor time to planning or evaluation.

38 Future studies might seek a better control variable than the Statistics Readiness Test. Variables such as SAT scores, GPA, and previous logic courses might be better indicators of statistics readiness than the simple algebra test used here. Future research into the effects of cooperative learning in college classrooms might better tease out these effects by more stringently controlling for confounding variables. For example, instructor interaction with students as they cooperate in groups could be limited. To control for the possibility that students in the cooperative group benefited from working more assignments than the lecture group, all assignments could be collected and scored for credit. More careful monitoring of time spent in study and completing practice items could control for differences between control and treatment classes. Finally, future research would benefit from the inclusion of more classes of each instructional type.

39 While it is important to attempt to further isolate the possible effects of cooperative learning, it is also important to note that the true test of an instructional practice is how it works for students (and instructors) in authentic classroom settings. This study gives evidence that cooperative learning can be a valuable asset in the college classroom.

Acknowledgments

I would like to thank Dr. Bandalos and Dr. Impara for advising me on this and other projects. I would also like to acknowledge the reviewers whose suggestions greatly improved this article.


Appendix

A.1 Sample Questions from Test 2

X1     Y1     X2     Y3
9      2      8      9
8      2      7      8
4      6      5      8
8      1      3      7
2      7      8      6
1      8      9      5
9      2      6      2


Describe the correlation between X1 and Y1, in terms of
direction and magnitude.

negative and moderate
negative and strong
positive and weak
positive and strong


Which relationship would give the most accurate predictions?

X1 Y1
X2 Y3
X1 X3
Y1 Y3

A.2 Sample Questions from the Final Examination

List ways that a researcher can increase the statistical
power of her study.


A researcher rejects the null hypothesis.  What statistical
error is possible?


A researcher wants to know if men and women prefer different
automobiles.  What statistical test could he use to answer
his question?

A.3 Sample Questions from the Statistics Readiness Test

What is the square root of 81?


Solve for x:

5 + x - 7 = 8

5 + x^2 + 2x = 8


Solve for y, when x = 5:

6 + 6x = y


References

Fantuzzo, J. W., King, J., and Heller, L. R. (1992), "Effects of Reciprocal Peer Tutoring on Mathematics and School Adjustment: A Component Analysis," Journal of Educational Psychology, 84, 331-339.

Garfield, J. (1993), "Teaching Statistics Using Small-Group Cooperative Learning," Journal of Statistics Education [Online], 1(1). (http://www.amstat.org/publications/jse/v1n1/garfield.html)

Jones, L. (1991), Using Cooperative Learning to Teach Statistics, Research Report 91-2, L. L. Thurstone Psychometric Laboratory, University of North Carolina.

Johnson, D. W., and Johnson, R. (1975), Learning Together and Alone: Cooperation, Competition, and Individualization, Englewood Cliffs, NJ: Prentice-Hall.

Johnson, R. T., and Johnson, D. W. (1985), "Student-Student Interaction: Ignored but Powerful," Journal of Teacher Education, 34(36), 22-26.

----- (1986a), "Action Research: Cooperative Learning in the Science Classroom," Science and Children, 24(2), 31-32.

----- (1986b), Encouraging Student/Student Interaction. Research Matters ... to the Science Teacher, ERIC Document ED266960, United States: National Association for Research in Science Teaching.

Johnson, D., Johnson, R., and Smith, K. (1991), Cooperative Learning : Increasing College Faculty Instructional Productivity, ASHE-ERIC Higher Education Report 4, Washington, D. C.: George Washington University.

Keeler, C. M., and Steinhorst, R. K. (1995), "Using Small Groups to Promote Active Learning in the Introductory Statistics Course: A Report from the Field," Journal of Statistics Education [Online], 3(2). (http://www.amstat.org/publications/jse/v3n2/keeler.html)

National Council of Teachers of Mathematics (1989), Curriculum and Evaluation Standards for School Mathematics, Reston, VA: NCTM.

----- (1991), Professional Standards for Teaching Mathematics, Reston, VA: NCTM.

National Research Council (1989), Everybody Counts: A Report to the Nation on the Future of Mathematics Education, Washington, D.C.: National Academy Press.

Pressley, M., and McCormick, C. B. (1995), Cognition, Teaching and Assessment, New York: HarperCollins.

Slavin, R. (1985), "Team-Assisted Individualization: Combining Cooperative Learning and Individualized Instruction in Mathematics," in Learning to Cooperate, Cooperating to Learn, eds. R. Slavin, S. Sharon, S. Kagan, R. H. Lazarowitz, C. Webb, and R. Schmuck, New York: Plenum, pp. 177-209.

Vygotsky, L. S. (1978), Mind in Society: The Development of Higher Psychological Processes, Cambridge, MA: Harvard University Press.

Wood, S., Bruner, J. S., and Ross, G. (1976), "The Role of Tutoring in Problem Solving," Journal of Child Psychology and Psychiatry, 17, 89-100.


Gerald Giraud
Box 251
Adams, NE 68503

ggiraud@unlgrad1.unl.edu


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